Engaging students: Graphing an ellipse

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Peter Buhler. His topic, from Precalculus: graphing an ellipse.

green lineHow could you as a teacher create an activity or project that involves your topic?

One project that could be assigned to students during the unit on conic sections could be to challenge students to either find or make an ellipse. This could be with a household object, a computer simulated object, or it could be something such as the movement of the planets around the sun. Students would be expected to visually display their object(s) of choice, as well as provide an equation for the ellipse. For example, if the student chose to use a deflated basketball or football, students would use the actual units found when measuring the object and then create an equation for that ellipse. Of course, students would also be expected to graph the ellipse using the appropriate equation, and then check the graph with the actual object (if possible). This project would allow students to be creative in choosing something of ellipse form, and would allow them to further explore the graphing and equation-building of an ellipse.


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How can this topic be used in your students’ future courses in mathematics or science?

While graphing an ellipse is a topic within the Pre-Calculus curriculum, it also has applications within other topics as well. One of these is the unit circle, which is also taught in most Pre-Calculus courses. The unit circle is simply an ellipse where both major and minor axis are of length 1, as well as the center at (0,0). Students can be encouraged to draw comparisons between the two topics. Not only can they rewrite the equation of an ellipse to fit the unit circle, but students can also use the distance formula to calculate sine and cosine values on the unit circle. They can then use the distance formula on various forms of an ellipse, and compare and contrast between the two.

Later on in a students’ mathematical career, some students may encounter ellipse used in three dimensions in Calculus III, in an engineering course, or even in an astronomy course. Ellipses have many applications, and students may benefit from you (as the teacher) perhaps mentioning some of these applications when going over the unit on conic sections.



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How has this topic appeared in high culture?

One particularly intriguing application of an ellipse (among many applications) is in the design of a whispering gallery. This is essentially a piece of architecture that is designed in the shape of an ellipse so that when someone is standing at one focus of the ellipse, they can clearly hear someone whispering from the exact location of the other focus. Some of examples of these “whispering rooms include St. Paul’s Cathedral, the Echo Wall in Beijing, and in the U.S. Capitol building. It has been commonly noted that President John Quincy Adams would eavesdrop on others while standing in the Capitol, simply due to the physics of sound waves traveling inside an ellipse shaped building.

On a more personal business, I can remember multiple visits to the Science Museum in Fair Park, where various forms of sciences were displayed in formats that children (and adults!) could interact with. There was one exhibit that was set up for several years that also incorporated this ellipse-shaped architecture. I remember it clearly, due to the fact that I was so fascinated with how I could stand 30 yards from someone and be able to hear their whisper clearly. This could also be a class project or even a class trip that would allow students to hypothesize why this works the way it does. It can be noted that this would work for both Physics and Math classes, as it has applications to both.


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